Earth-Sun geometry
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Position of the equinoxes, solstices, aphelion, and perihelion relative to the Earth's orbit around the Sun.
Contents
Earth Rotation and Revolution
The term 'Earth rotation' refers to the spinning of the Earth on its axis. One rotation takes exactly twenty-four hours and is called a mean solar day. If you could look down at the Earth's North Pole from space you would notice that the direction of rotation is counterclockwise. The opposite is true if you viewed the Earth from the South Pole.
The orbit of the Earth around the sun is called Earth revolution. This celestial motion takes 365 1/4 days to complete one cycle. Further, the Earth's orbit around the sun is not circular, but elliptical (see Figure 1). An elliptical orbit causes the Earth's distance from the sun to vary annually. However, this phenomenon does not cause the seasons! This annual variation in the distance from the sun does influence the amount of solar radiation intercepted by the Earth by approximately 6%. On January 3rd, perihelion, the Earth is closest to the sun (147.5 million kilometers) (Figure 1). The Earth is farthest from the sun on July 4th, or aphelion (Figure 1). The average distance of the Earth from the sun over a one year period is 150 million kilometers.
Tilt of the Earth's Axis
Figure 2: Annual change in the position of the Earth in its revolution around the sun. (Source: PhysicalGeography.net) The Earth's axis is not perpendicular to the plane of the ecliptic, but inclined at a fixed angle of 23.5°. Moreover, the northern end of the Earth's axis always points to the same place in space (North Star). Figure 3 shows an animation of the Earth revolving around the sun. In this animation the Earth's axis is colored red. Note that the angle of the Earth's axis in relation to the plane of the ecliptic remains unchanged. However, the relative position of the Earth's axis to the sun does change during this cycle (Figure 2). This circumstance is responsible for the annual changes in the height of the sun above the horizon. It also causes the seasons, by controlling the intensity and duration of sunlight received by locations on the Earth.
Figure 3: Earth revolution animation. (Source: PhysicalGeography.net) Figure 2 shows the annual change in the position of the Earth in its revolution around the sun. In the graphic, we are viewing the Earth from a position in space that is above the North Pole (yellow dot) at the summer solstice, the winter solstice, and the two equinoxes. Note how the position of the North Pole on the Earth's surface does not change. However, its position relative to the sun does change and this shift is responsible for the seasons. The red circle on each of the Earths represents the Arctic Circle (66.5° N). During the summer solstice, the area above the Arctic Circle is experiencing 24 hours of daylight because the North Pole is tilted 23.5° toward the sun. The Arctic Circle experiences 24 hours of night when the North Pole is tilted 23.5° away from the sun in the winter solstice. During the two equinoxes, the circle of illumination cuts through the polar axis and all locations on the Earth experience 12 hours of day and night.
On June 21 or 22, the summer solstice, the Earth is positioned in its orbit so that the North Pole is leaning 23.5° toward the sun (Figures 2, 4). During the summer solstice, all locations North of the equator have day lengths greater than twelve hours, while all locations South of the equator have day lengths less than twelve hours. On December 21 or 22, the winter solstice, the Earth is positioned so that the South Pole is leaning 23.5° toward the sun (Figures 2, 4). During the winter solstice, all locations North of the equator have day lengths less than twelve hours, while all locations South of the equator have day lengths greater than twelve hours.
On September 22 or 23, the autumnal equinox, neither pole is tilted toward the sun (Figures 2, 5). March 20 or 21 marks the arrival of the spring or vernal equinox when once again the poles are not tilted toward the sun. Day lengths on both of these days, regardless of latitude, are exactly 12 hours.
Figure 4: During the summer solstice the Earth's North Pole is tilted 23.5° towards the sun relative to the circle of illumination. This phenomenon keeps all places above a latitude of 66.5° N in 24 hours of sunlight, while locations below a latitude of 66.5° S are in darkness. The North Pole is tilted 23.5° away from the sun relative to the circle of illumination during the winter solstice. On this date, all places above a latitude of 66.5° N are now in darkness, while locations below a latitude of 66.5° S receive 24 hours of daylight. (Source: PhysicalGeography.net) 
Figure 5: During the equinoxes, the axis of the Earth is not tilted toward or away from the sun and the circle of illumination cuts through the poles. This situation does not suggest that the 23.5° tilt of the Earth no longer exists. The vantage point of this graphic shows that the Earth's axis is inclined 23.5° toward the viewer for both dates (see Figure 2). The red circles shown in the graphic are the Arctic Circle. (Source: PhysicalGeography.net) Axis Tilt and Solar Altitude
The annual change in the relative position of the Earth's axis in relationship to the sun causes the height of the sun (solar altitude) to vary in our skies. The total variation in maximum solar altitude for any location on the Earth over a one year period is 47° (2 x 23.5 = 47). For example, at 50° North, maximum solar altitude varies from 63.5° on the summer solstice to 16.5° on the winter solstice (Figure 6). Maximum solar height at the equator goes from 66.5° above the northern end of the horizon during the summer solstice, to directly overhead on the fall equinox, and then down to 66.5° above the southern end of the horizon during the summer solstice (Figure 7).
Figure 6: Variations in solar altitude at Solar noon for 50° North during the summer solstice, equinox, and winter solstice. (Source: PhysicalGeography.net) 
Figure 7: Variations in solar altitude at solar noon for the equator during the summer solstice, equinox, and winter solstice. (Source: PhysicalGeography.net) The location on the Earth where the sun is directly overhead at solar noon is known as the subsolar point. The subsolar point occurs on the equator during the equinoxes (Figure 8). During the summer solstice, the subsolar point moves to the Tropic of Cancer because at this time the North Pole is tilted 23.5° toward the sun. The subsolar point is located at the Tropic of Capricorn on the winter solstice. On this date, the South Pole is now tilted toward the sun (Figures 2 and 4).
Figure 8: Relationship of maximum sun height to Latitude for the equinox (left) and summer solstice (right). (Source: PhysicalGeography.net) Figure 8 shows the relationship of maximum sun height to the latitude for the equinox (left) and summer solstice (right). The red values on the right of the globes are maximum solar altitudes at solar noon. Black numbers on the left indicate the location of the equator, Tropic of Cancer (23.5° N), Tropic of Capricorn (23.5° S), Arctic Circle (66.5° N), and the Antarctic Circle (66.5° S). The location of the North and South Poles are also identified. During the equinox, the equator is the location on the Earth with a sun angle of 90° for solar noon. Note how maximum sun height declines with latitude as you move away from the equator. For each degree of latitude traveled, maximum sun height decreases by the same amount. At equinox, you can also calculate the noon angle by subtracting the location's latitude from 90. During the summer solstice, the sun is now directly overhead at the Tropic of Cancer. All locations above this location have maximum sun heights that are 23.5° higher from the equinox situation. Places above the Arctic Circle are in 24 hours of daylight. Below the Tropic of Cancer the noon angle of the sun drops one degree in height for each degree of latitude traveled. At the Antarctic Circle, maximum sun height becomes 0° and locations south of this point on the Earth are in 24 hours of darkness.
The following table describes the changes in solar altitude at solar noon for the two solstices and equinoxes. All measurements are in degrees (horizon has 180 degrees from True North to True South) and are measured from either True North or True South (whatever is closer).
| Table 1: Maximum sun altitudes for selected latitudes during the two solstices and equinoxes. | ||||
|---|---|---|---|---|
| Location's Latitude |
Vernal Equinox March 20/21 |
Summer Solstice June 21/22 |
Autumnal Equinox September 22/23 |
Winter Solstice December 21/22 |
| 90° N | 0° | 23.5° | 0° | - 23.5° |
| 70° N | 20° | 43.5° | 20° | -3.5° |
| 66.5° N | 23.5° | 47° | 23.5° | 0° |
| 60° N | 30° | 53.5° | 30° | 6.5° |
| 50° N | 40° | 63.5° | 40° | 16.5° |
| 23.5° N | 66.5° | 90° | 66.5° | 43° |
| 0° | 90° | 66.5° | 90° | 66.5° |
| 23.5° S | 66.5° | 43° | 66.5° | 90° |
| 50° S | 40° | 16.5° | 40° | 63.5° |
| 60° S | 30° | 6.5° | 30° | 53.5° |
| 66.5° S | 23.5° | 0° | 23.5° | 47° |
| 70° S | 20° | -3.5° | 20° | 43.5° |
| 90° S | 0° | - 23.5° | 0° | 23.5° |
Finally, the altitude of the sun at solar noon can also be calculated with the following simple equation:
In this equation, L is the latitude of the location in degrees and D is the declination. The equation is simplified to A = 90 - L if sun angle determinations are being made for either equinox date. If the sun angle determination is for a solstice date, declination (D) is added to latitude (L) if the location is experiencing summer (northern latitudes = summer solstice; southern latitudes = winter solstice) and subtracted from latitude (L) if the location is experiencing winter (northern latitudes = winter solstice; southern latitudes = summer solstice). All answers from this equation are given relative to True North for southern latitudes and True South for northern latitudes. For our purposes only the declinations of the two solstices and two equinoxes are important. These values are: Summer Solstice D=23.5, Winter Solstice D=23.5, Autumnal Equinox D=0, and Vernal Equinox D=0. When using the above equation in tropical latitudes, sun altitude values greater than 90° may occur for some calculations. When this occurs, the noonday sun is actually behind you when looking towards the equator. Under these circumstances, sun altitude should be recalculated as follows:
